Problem: You have found the following ages (in years) of 6 porcupines. Those porcupines were randomly selected from the 47 porcupines at your local zoo: $ 11,\enspace 13,\enspace 6,\enspace 1,\enspace 3,\enspace 8$ Based on your sample, what is the average age of the porcupines? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 47 porcupines, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{11 + 13 + 6 + 1 + 3 + 8}{{6}} = {7\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {16} + {36} + {1} + {36} + {16} + {1}} {{6 - 1}} $ {s^2} = \dfrac{{106}}{{5}} = {21.2\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{21.2\text{ years}^2}} = {4.6\text{ years}} $ We can estimate that the average porcupine at the zoo is 7 years old. There is also a standard deviation of 4.6 years.